where the two-component identity matrix \(\tau ^{0}\)  and the standard Pauli
matrices \(\tau ^{i}\)  act on the valley indices (\(K,-K\) ), the two-component
Pauli matrices \((\sigma ^{0},\sigma ^{i})\)  act in sublattice space \((A,B)\) .
In the free Dirac Lagrangian, the four-component Dirac spinor is defined as \(\Psi =(c_{AK},c_{BK},c_{A-K},c_{B-K})^{\text{T}}\) . In the vicinity of Dirac
points, then the Bloch Hamiltonian reads \(\mathcal {H}=\gamma ^{0}\gamma ^{i}k_{i}\)  with the reduced Planck constant \(\hbar =1\)[1]}. There are two
matrices anticommute with all \(\gamma ^{\mu }\)  matrices
\(\gamma ^{3}=\tau ^{1}\otimes i\sigma ^{2},\gamma ^{5}=\tau ^{2}\otimes i\sigma ^{2}.\) 
